Relation between Molal Volumes and Molal Compressibilities
2367
Relation between Molal Volumes and Molal Compressibilities from the Viewpoint of the Scaled-Particle Theory. Prediction of the Apparent Molal Compressibilities of Transfer N. Desrosiers Bpartement de Chimie, Universite de Sherbrooke,Province of Quebec, JIK 2RI Canada
and M. Lucas' Bpartement de G n i e Radioactif, Commissariat a I'Energie Atomique, 92260 Fontenay-aux-Roses, France (Received February 15, 1974) Publication costs assisted by the Commissariat a I'Energie Atomique
By means of the scaled-particle theory. the compressibility of transfer from H20 to 3 m urea and t o D20 can be calculated for various salts provided one knows the compressibility in water and the molal volume of salt in 3 m urea or in D20. T h e fit between experimental and calculated values is good.
The partial molal volumes and partial molal compressibilities of solutes in water and aqueous solutions are considered important thermodynamic quantities, whose knowledge hopefully gives information about the structural influence of the solute upon water structure.' They are usually considered as independent quantities. However. the scaled-particle theory gives a relation between these two quantities and the solute hard-sphere diameter. In this paper, the relation is given and it is shown that the compressibility of transfer from H20 to 3 m urea and from H20 to D20 can be calculated for various salts and, in many cases, within the experimental error. Theory The coefficient of isothermal compressibility 8 of a mixture of hard spheres of different diameters is given by eq 16 of ref 2
1 -
13
6kT [S(l a ( l - 2)j
z)?
+ 6XY(1 - Z) 9x3
+
4
Z,
= 130 -+ A m
BVZ~/~
(3
- Clo@Ovand lOOOB = SI( - 30Sb.. Then
z2x8- 4x%] (1)
rrS 55.51aP nup + w ( b P -c CP) 6 55.512:0H,20+ nd, + mdvs
-[
13
where IOOOA = @OK
+
In the case of a mixture of water, urea, and salt, we give the definition +
found a value of 2.76 A. Rowlinson? has found 2.73 A. Pierottij has found, by different methods, a water hardsphere diameter of 2.75 A, which is close to the oxygenoxygen intermolecular distance in ice (2.76 A).s Usually, in studies dealing with aqueous solutions, a hard-sphere diameter of 2.76 8, for water is used.6 Therefore we prefer t o use the scaled-particle theory in its simplest form that is by using only the hard-sphere part and neglecting the interaction terms. Also, the possibility exists that the interaction forces are somewhat taken into account in the case of the compressibilities by the fact that we introduce in the equations the experimental volume of the solutes in the studied solvent. The concentration dependence of the coefficient of isothermal compressibility can be expressed as7
I=
X, I;, o r S
At infinite dilution? we have
(5) and then
(2)
where p = 3 for t,p = 2 for X,p = 1 for Y,and, finally, p = 0 for S In these equations, a , u , b, and c are respectively the hard-sphere diameters of water, urea, and anion and cation of the salt, n and m are respectively the urea and salt molalities, @ ~ is~theopure water molal volume, 4\,uis the apparent molal volume of urea in water, and is the apparent molal volume of salt in water ( n = 0) or of salt in aqueous urea solution ( n # 0). In their eq 16 Tiepel and Gubbins2 included terms dealing with interaction forces. Those terms, approximate in nature, when added t o complete eq 1 lead to a hard-sphere diameter of 2.98 A for water (ref 2, Table III). On the other hand, when those interaction terms are neglected, then eq 1 vields a hard-sphere diameter of water of 2 . i 4 A. A good fit bf the calculated second virial coefficients to the experimental values has been achieved by Stockmayer,3 who has
Also, taking into account that
(7)
(E) x am m-0
FoH2,do - 1000 A,, etc.
where
A, =
ap
v+cp + 0.018018nuP
-
The Journai of Physical Chemistry. Vol. 76. No. 23. 1974
N. Desrosiers and M. Lucas
2368
and d o is the pure water density, it is then possible to derive, from eq l, that
salt solution a t aquamolality m'; Bo and do are respectively the coefficient of isothermal compressibility and density of the solvent (either pure water or aqueous urea). W ~ j j lis the weight of 55.51 mol of solvent or mixed solvent; i.e. w5505i =
55. 51iv, + ?lhf, (55.51 + n)/55.51
(15)
where M, and Mu are respectively water and urea molecular weights. In the limit of infinite dilution
Finally
4z3X3 - 16z2X3 + 36X3z (1 - 2)s - 6XY (1 - 2 ) 3
+
9X3
+
+
with m = rn' (55.51 n)/55.51. The knowledge of @Ov,and $OK for a given salt in pure water make it possible to compute b and c for a salt provided that a for water is known. This parameter is calculated from the experimental compressibility of pure water of the equation
z 2 X 3 - 4zX3
(1 -
2)4
When there is no urea ( n = 0 )
1 - - RT
and 2
= Ni7a3/6F3
do H20
+
~ 3 ( 1- 2)4
2)'
A3(22
3A2z(1 - 2)(2 + 7 2 - 42' + z3) + 142' + 16z3 - 102' + 2z5)] (13) @,qof
a
= 6 $ v s f w5,.51(P - P o ) / m ' d o (14) taken from ref 8. J is the coefficient of isothermal compressibility of the @K
The Journal of Physical Chemistry. Vol. 78. No. 23. 1974
22)'
- 4z3
(1 -
+
z4
(18)
ZY
Discussion
+
The apparent molal isothermal compressibility salt is given by the relation
+
u is found to be equal to 4.408 8,
18z3 4z4 + a3(1 - 2)4 &(I - 2)4
And finally, since all a 3 vanish
6A1z(l -
(1
taken from ref 9. In this, z = T N ~ ~ I ~ atV 25O, O ~poH20 , ~ = 45.25 X bar-' and P/OH~O = 18.057 cm3 Then, a is found to be equal to 2.74 A, which is not much different from the value 2.76 8, computed by Pierotti with another method. For urea in water a t infinite dilution, @OK = 0.90 x bar-' cm3 mol-' and @Ov, = 44.20 cm3 Then, from eq 5 , with
Then eq 10 reduces to
2 2
- I'OH20
The hard-sphere diameters of monovalent ions are given in Table I. They have been computed from the experimental 4~ of salts in water at 0.1 rn,*and experimental $O\., of salts given in ref 11 ( @ O K should have been taken instead of 4~ a t a finite salt molality, but the rule of extrapolation to zero salt concentration is not known for the @ K ) . In Table I are also given crystallographic ionic diameters for the same ions. The comparison between the two sets of valws shows that, except for F-, the sets of ionic diameters are very similar. This implies that the ionic response to a variation of pressure is the same as if the water of hydration could be considered as separate particles. Once a, b, c, and u are known, the $ O K for salts in 3 rn urea can be computed, if @ o ~ ~for g salts in urea are known. In fact only $ v s a t salt aquamolality 0.1 rn was available. at Hence @ O v , in 3 rn urea has been computed from 0.1 m in 3 m urea and @ v sfor the same salt in pure water assuming that the A@btr(H20 3 rn urea) is salt concentration independent. However it has been shown that A@v,,(HzO 3 rn urea) shows only small salt concentration dependence,given by the equation8
-
AdVs (H20 ti-
-
-
4
~
7
3 m urea) = 2.14 - 0.39 rrz
-
where rn is the salt aquamolality. Thus @v.(m = 0.1) must be very nearly equal to @Ov,. [ 1 @ ~ , , ( H 2 0 3 m urea)
~
Relation between Molal Volumes and Molal Compressibilities
2369
TABLE I: I o n i c D i a m e t e r s of Monovalent Ions (Angstroms) Li Computed Crystallographic b
+
1.37" 1.37
Na -
Kf
c s-
F-
c1-
Br-
I-
1.836 1.94
2.658 2.66
3.329 3.34
3.266 2.66
3,792 3.62
3.948 3.92
4,222 4.32
* "Handbook
Assumed arbitrarily in the initial calculation. lishing Co., Cleveland, Ohio, 1964-1965. 0
of Chemistry and Physics," 45th ed, Chemical Rubber Pub-
-
TABLE 11: C o m p a r i s o n of Experimental and Calculated Apparent Molal I s o t h e r m a l Compressibilities, oL x 104 (bar-' c m 3mol - ), and T r a n s f e r F u n c t i o n s A + K ~ H , O 3 m u r e a or D?O) X l o 4 ( b a r - l c m 3 mol-') at 25" Salt Solvent
LiCl
NaCl
KCl
CSCl
NaI
NaBr
NaF
H,0r4 exptl 3 m urea" exptl 3 m urea5 computed &bx(HZO + 3 m urea). exptl A+K(H?O 3 m urea)b computed A+K.(H?O + D,O)t exptl A$K ( H 2 0 + D?O)* computed
- 38 -26.7 -28.8 11.3 9.2
-45.4 -33.0 -35.2 12.4 10.2 -4.4 -3.2
-38.7 -26.7 -28.6 12 .o 10.1 -2.8 -3.0
-31.3 -21.7 -21.4 9.6 9.9 -1.7 -1.1
-25.7 -14.7 -16.3 11 .o 9.4
-37.2 -25.9 -27.8 11.3 9.4
-67.6 -54.4 -56.9 13.2 10.7 -5.5 -4.7
-f
Values from ref 8. Values from present work. Adiabatic values from J. G. Mathieson and B. E. Conway, J. Chem. SOC.,Faraday Trans. I, in press. The necessary Spot.have been taken from ref 11. I+
I_
cient account of attractive potential between particles is also shows a small concentration dependence which is given by A @ K , = , 13.30 - 2.6m.&] taken, if experimental volumes are included in the equaTable I1 shows the calculated $ K in aqueous 3 m urea. tion. Experimental @K values of salts 0.1 m in 3 m urea are also Acknowledgments. Helpful suggestions from Professor J. given. The agreement with calculated values of 4~ is good, E. Desnoyers are gratefully acknowledged. N. D. is also the difference in many cases being smaller than 6%. The grateful to the National Research Council of Canada and to computed and experimental - 1 @ ~ , , ( H 2 0 3 m urea) the Cooperation Franco-QuBbBcoise for the award of a values are also given (experimental values are considered scholarship. Helpful comments from Dr. Jean-Pierre Morel bar-' cm-3 mol-'). Table I1 also inexact within 1 X cludes the calculated 1 4 ~ , , ( H ~ 0D20) values. They are have been appreciated. compared to experimental 1 @ K values. Although this comparison is weakened by the fact that the calculated 1 4 ~ References and Notes values are isothermal, whereas in the case H2O D20 the ( 1 ) J. E. Desnoyers and P. R. Philip. Can. J. Cbem., 50, 1094 (1972). experimental ones are adiabatic, the sign and order of mag(2) E. W. Tiepel and K. E. Gubbins, J. Pbys. Cbem.,76, 3047 (1972). (3) W. H. Stockmayer. J. Chem. Phys., 9, 398 (1941). nitude are probably correct. [3O(D20) and VO(D20) are (4) J. S.Rowlinson, Trans. Faraday SOC.,47, 120 (1949). from ref IO.] ( 5 ) R. A . Pierotti. J. Pbys. Cbem.. 69, 281 (1965). (6) H. L. Friedman in "Water, a Comprehensive Treatise," Vol. 3, F. Since experimental volumes of the salts are taken, strucFranks, Ed., Plenum Press, New York, N. Y.. 1973, p 49. tural information on the influence of the salts upon water (7) H. S.Harned and B. B. Owen, "The Physical Chemistry of Electrolyte Solutions," Reinhold, New York. N. Y., 1958, Chapter 8. and 3 m urea structure are implicitly present in the calcu(8) N. Desrosiers, G. Perron, J. G. Mathieson, B. E. Conway, and J. E. Deslated @K. One may also guess that the success of the scalednoyers, J. Solution Chern., in press. particle theory in this particular case comes from the fact (9) T. Boublik, J. Chem. Phys., 53, 471 (1970). (10) G. S. Kell in "Water, a Comprehensive Treatise," Vol. 1. F. Franks, Ed., that the repulsive potential between particles in a solution Plenum Press, New York, N. Y.. 1972, p 384. plays a major role in the determination of the 6~ (and the ( 1 1) J.-L. Fortier, P. R. Philip, and J. E. Desnoyers. J. Solution Cbem., in scaled-particle theory takes it into account), whereas suffipress.
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The Journal of Physical Chemistry. Vol. 78. No. 23. 1974
